Klein–Kramers equation

In physics and mathematics, the Klein–Kramers equation or sometimes referred as Kramers–Chandrasekhar equation is a partial differential equation that describes the probability density function $f (r, p, t)$ of a Brownian particle in phase space $(r, p)$ . It is a special case of the Fokker–Planck equation.

In one spatial dimension, $f$ is a function of three independent variables: the scalars $x$ , $p$ , and $t$ . In this case, the Klein–Kramers equation is

{\frac {\partial f}{\partial t}}+{\frac {p}{m}}{\frac {\partial f}{\partial x}}=\xi {\frac {\partial }{\partial p}}\left(p\,f\right)+{\frac {\partial }{\partial p}}\left({\frac {dV}{dx}}\,f\right)+m\xi k_{\mathrm {B} }T\,{\frac {\partial ^{2}f}{\partial p^{2}}}

where $V (x)$ is the external potential, $m$ is the particle mass, $ξ$ is the friction (drag) coefficient, $T$ is the temperature, and $k B$ is the Boltzmann constant. In $d$ spatial dimensions, the equation is

{\frac {\partial f}{\partial t}}+{\frac {1}{m}}\mathbf {p} \cdot \nabla _{\mathbf {r} }f=\xi \nabla _{\mathbf {p} }\cdot \left(\mathbf {p} \,f\right)+\nabla _{\mathbf {p} }\cdot \left(\nabla V(\mathbf {r} )\,f\right)+m\xi k_{\mathrm {B} }T\,\nabla _{\mathbf {p} }^{2}f

Here $\nabla _{\mathbf {r} }$ and $\nabla _{\mathbf {p} }$ are the gradient operator with respect to $r$ and $p$ , and $\nabla _{\mathbf {p} }^{2}$ is the Laplacian with respect to $p$ .

The fractional Klein-Kramers equation is a generalization that incorporates anomalous diffusion by way of fractional calculus.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.